Saturday, August 14, 2010

The Econ Geek's Guide to Deal or No Deal: an Empirical Study

The game show "Deal or No Deal" is an econ geek's dream. Not only is it a thrilling spectacle for game show lovers, it is also a laboratory for studying human risk-taking behavior. For those who don't know the rules, on the show there are 26 numbered briefcases, each with a tag inside, showing an amount of money. The amount of money in each case ranges from 1 penny to 1 million dollars. The contestant first chooses one of the cases to take into possession, and then through the rest of the game, eliminates cases from the remaining 25, starting with 6 cases at once, then 5 then 4 then 3 then 2 then 1 at a time until all but the contestant's case is gone. As each case is eliminated, the amount it contains is exposed, thus letting the contestant know what amount is not in her own case. If the contestant eliminates all of the 25 cases, she walks away with the amount of money in the initially chosen case. The twist is that there is a "banker" on the show who after each round of elimination, offers the contestant an amount of money to stop playing. Because, superstitions aside, the choice of eliminating one numbered case over another does not matter, the only pertinent decision in the game is whether to take the deal or keep playing, (which makes the title of the show particularly fitting).
As the contestant continues to eliminate from the 25 cases, by inference, she gets a better idea of what is in her own case, and so does the banker. So if the contestant eliminates the case that has the penny, that's a good thing, because it means that the personal case doesn't contain the penny. If the contestant eliminates the million dollars, she knows that her personal case doesn't contain the million, and this is a bad thing.
For years I have watched this show, and wondered "how does the banker choose the amounts of each offer?" After quantitatively studying this (albeit with a limited sample of 64 offers from 9 complete games) I think I have come close to answering this question.
To understand how the banker makes his offers, there's one key mathematical concept to keep in mind: expected value. Expected value is equal to the sum of the values of all possible outcomes multiplied by their respective probabilities. It is the average amount of money per person that a large group of people would win on this game if they never took deals.
When one starts the game, the dollar values of the cases are as follows:
$0.01, $1, $5, $10, $25, $50, $75, $100, $200, $300, $400, $500, $750, $1000, $5000, $10000, $25000, $50000, $75000, $100000, $200000, $300000, $400000, $500000, $750000, $1000000
And the probabilities of each outcome are equally likely: 1/26= (approximately) 0.03846.
So, the expected value, (in this case also just the average of all values because the probabilities are the same), is equal to:

$0.01/26 + $1/26 + $5/26 + $10/26 + $25/26 + $50/26 + $75/26 + $100/26 + $200/26, +$300/26 + $400/26 + $500/26 + $750/26 + $1000/26 + $5000/26 + $10000/26 + $25000/26 + $50000/26 + $75000/26 + $100000/26 + $200000/26 + $300000/26 + $400000/26 + $500000/26 + $750000/26 + $1000000/26 = $131,477
So, when you start the game, the expected value of your personal case, before any of the remaining cases are eliminated, is $131,477. What if, before you even started playing, the banker offered you $80,000 to not play the game at all? Would you take the deal? On average, taking this deal would get contestants much less than playing through all the way. But for reasons I shall explain later in this article, the banker usually makes offers that, just like this one, are significantly lower than the expected value of the case, and despite the low offers, very few contestants actually play through all the cases.
As one plays the game, and eliminates cases, the expected value of what's in one's personal case changes. Lets say that the contestant has eliminated 24 of the cases, leaving just the personal case and one other case in play, and that the only two possible values remaining are $0.01 and $1,000,000. Because the probability of either outcome is 1/2, the expected value of the contestant's personal case would be equal to:
$0.01/2 + $1,000,000/2 = $500,000
What if at this point in the game, the banker offers a deal for $250,000? Ask yourself: if you had a choice here between choosing your case, which might have a million dollars in it or might have a penny, or taking a deal for $250,000, what would you do?
Personally I would take the $250,000. This illustrates an important concept in economics: risk aversion. I am risk averse in this situation because I would choose a certain reward over an uncertain one, even if the expected value of the reward in the uncertain event is greater than the certain reward.
So in Deal or no Deal, the banker always wants the contestant to take the deal, right? Wrong. If contestants took the first or second deals there wouldn't be much of a show, and the network would need to bring on more contestants, and thus give away more prizes, to fill airtime. On top of this, the show tends to get more interesting as it goes along and people make the riskier decisions. For these two reasons, one related to the costs of broadcasting the show, and the other related to the benefits of having a more interesting show, there is an incentive to make lower offers to the contestants in early rounds to get them to play for longer.
The quality of an offer can actually be quantified. There's a measurement I use to find the quality of a deal relative to what cases are still in play. I call it the "Offer Quality Ratio", and here it is:
offer quality ratio = (offer amount)/(expected value of remaining cases at time of offer)

So lets say there are the following cases left on the board (by the way, this is from an actual episode): $1, $100, $50000, and $100000
and the contestant gets an offer of $27000.
The expected value is = $1/4 + $100/4 + $50000/4 +$100000/4 = $37525.25
So the Offer Quality Ratio = $27000/$37525.25 = 0.7195
In other words, the offer is around 72% of the expected value of playing to the end.

After the challenging task of watching TV for hours, I have collected data on 64 offers from 9 complete Deal or No Deal games. My results show that as contestants play the game, they tend to get rewarded with higher quality deals relative to what cases are still in play. From these 9 games, here are the average Offer Quality Ratios for offers in each round:

First offer: 28.9% of expected value, (Standard Deviation 15.8%)
Second offer: 42.8% of expected value, (Standard Deviation 15.0%)
Third offer: 47.7% of expected value, (Standard Deviation 13.6%)
Fourth offer: 54.8% of expected value, (Standard Deviation 12.2%)
Fifth offer: 65.1% of expected value, (Standard Deviation 17.8%)
Sixth offer: 65.0% of expected value, (Standard Deviation 16.1%)
Seventh offer: 84.0% of expected value, (Standard Deviation 17.3%)
Eighth offer: 90.6% of expected value, (Standard Deviation 18.3%)
Ninth offer: 97.6% of expected value (S. Dev 3% from a limited sample of just two offers)

Though I'm sure this study would benefit from a larger sample size, there are two conclusions I have drawn from it.
First, the quality of deals relative to what is on the board tends to rise as games progress. As you can see, the first offer tends to be incredibly low, and is usually not even worth considering.
Only the most risk averse contestants would take a first offer that's under 30% of expected value. As the game progresses, the managers of the show, weighing the costs of giving a bigger payout against the benefits of a more interesting show and more airtime per contestant, increase the quality of the offers.
Secondly, the standard deviation (the average amount of sample variation above or below the average value of all elements in the sample) of offers is significant, at around 15 to 20 Offer Quality percentage points, and remains rather constant throughout the game. This is either because there is an element of randomness built into the offer-determining formula used on the show, or there are hidden variables determining part of each offer. Maybe contestants are given a psych evaluation before the show that gives insight into their risk profiles? This would help the banker minimize payouts while maximizing the thrill of the show.

So, the moral of the story is, if you are ever on Deal or no Deal, fortune might favor you because of your boldness. Rather, I should say, the "banker" (and by that I mean the managers and producers of the show) might favor you with a good offer because you've made the show more interesting, and thus more profitable.

Sunday, August 8, 2010

Why Soccer is Less Popular in the U.S.

What is it about soccer that has stopped it from really taking off as a spectator sport in the United States? Could it be the low goal scoring? The constant change of possession? In my opinion, the answer has less to do with the aesthetics of the game than it does with economics (big surprise, right?). Allow me to explain.
To understand the market for televised soccer, one must first understand the economic quirks of the television market in general. Specifically, antenna television has the problem of being what economists call a "public good." A public good has two characteristics:
1) It is non-rivalrous in consumption, meaning one person's consumption of the good does not prevent others from using it. While goods like apples are rivalrous in consumption, meaning if you eat an apple, someone else cannot also eat that apple, one person's watching a television program does not stop anyone else from watching it on a separate TV.
2) It is non-exclusive in consumption, meaning nobody can be stopped from consuming the good if they want to consume it, making it impossible to collect money in exchange for consumption. While a grocery store can prevent the theft of its apples, a TV network broadcasting over airwaves cannot prevent anyone with a television from harnessing those airwaves to watch TV programs.

Cable and satellite broadcasts, however are excludable. These advancements have bypassed the problem of money collection for TV services. But things were different before the age of cable TV. In these early decades, there were two choices for financing television, either publicly funding it, or getting revenues from advertisers. The United States, unlike many other countries in the world, has relied primarily upon a private system of television funding, based on advertising. In the US, the advertiser became the real customer, paying for airtime, and the TV viewer was a bystander to the process. The commercial break became a necessity. In other countries however, governments stepped in to create networks like the BBC in Britain, that were funded by taxation, thus eliminating the need for commercial breaks.

But what does this have to do with soccer? Quite a lot really. The game of soccer is divided into two continuous 45 minute halves. Other than half-time there are no natural breaks in the game, like time-outs in American football or basketball, or breaks between 9 different innings as there are in baseball. This is a problem for broadcasters who depend on commercial breaks for their only source of revenue. For this reason, in an advertising based financing system, soccer games will tend to be chosen less than other programs. Why show a soccer game with only a few commercial breaks during half-time, when you can show a basketball game with numerous time-outs, breaks between quarters, and a half-time break? So antenna-televised soccer is not just a public good, but a public good that is resistant to advertising.
To explain why soccer is not so popular in the United States, my theory is, in recent decades when soccer became the world's most popular sport, its lack of exposure on US television played a role in its relative lack of popularity. Soccer haters may disagree, but the economic logic is sound. The pay-for cable and satellite sports networks that sprang up in the age of cable, or new forms of web based broadcasting, may eventually give soccer the exposure it needs to be on par with football, baseball and basketball. Just don't expect any of the traditional networks to broadcast a soccer game when there's a perfectly good basketball game to show.